L(s) = 1 | + (0.156 + 0.987i)2-s + (−0.987 − 0.156i)3-s + (−0.951 + 0.309i)4-s − i·6-s + (−0.453 − 0.891i)8-s + (0.951 + 0.309i)9-s + (1.29 − 1.51i)11-s + (0.987 − 0.156i)12-s + (0.809 − 0.587i)16-s + (0.763 − 0.0600i)17-s + (−0.156 + 0.987i)18-s + (−0.891 + 1.45i)19-s + (1.70 + 1.04i)22-s + (0.309 + 0.951i)24-s + (0.587 + 0.809i)25-s + ⋯ |
L(s) = 1 | + (0.156 + 0.987i)2-s + (−0.987 − 0.156i)3-s + (−0.951 + 0.309i)4-s − i·6-s + (−0.453 − 0.891i)8-s + (0.951 + 0.309i)9-s + (1.29 − 1.51i)11-s + (0.987 − 0.156i)12-s + (0.809 − 0.587i)16-s + (0.763 − 0.0600i)17-s + (−0.156 + 0.987i)18-s + (−0.891 + 1.45i)19-s + (1.70 + 1.04i)22-s + (0.309 + 0.951i)24-s + (0.587 + 0.809i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7856674231\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7856674231\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.156 - 0.987i)T \) |
| 3 | \( 1 + (0.987 + 0.156i)T \) |
| 41 | \( 1 + (-0.891 - 0.453i)T \) |
good | 5 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 + (-0.891 + 0.453i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 1.51i)T + (-0.156 - 0.987i)T^{2} \) |
| 13 | \( 1 + (0.453 - 0.891i)T^{2} \) |
| 17 | \( 1 + (-0.763 + 0.0600i)T + (0.987 - 0.156i)T^{2} \) |
| 19 | \( 1 + (0.891 - 1.45i)T + (-0.453 - 0.891i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.987 - 0.156i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + (-1.16 + 0.183i)T + (0.951 - 0.309i)T^{2} \) |
| 47 | \( 1 + (0.891 + 0.453i)T^{2} \) |
| 53 | \( 1 + (0.987 + 0.156i)T^{2} \) |
| 59 | \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 67 | \( 1 + (0.843 + 0.987i)T + (-0.156 + 0.987i)T^{2} \) |
| 71 | \( 1 + (-0.156 - 0.987i)T^{2} \) |
| 73 | \( 1 + (1.14 - 1.14i)T - iT^{2} \) |
| 79 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + 0.618iT - T^{2} \) |
| 89 | \( 1 + (0.355 + 1.47i)T + (-0.891 + 0.453i)T^{2} \) |
| 97 | \( 1 + (0.794 - 0.678i)T + (0.156 - 0.987i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.31475271346323334658071966757, −9.350104793932005031274092984975, −8.529595387200568567878865936077, −7.66918661005140623710276860253, −6.73001506508943683324170964683, −5.98129569787388407297429895727, −5.55911743783660018686463531392, −4.27324874024378164249199731561, −3.51401851994674569706916705819, −1.13537652305015730341117688843,
1.17885750635250723298046698876, 2.49636601451567028148191541970, 4.16493786341549593346928867407, 4.44412697876590107615237663801, 5.56275476445929065862916975460, 6.56029129695162513788863754401, 7.37663401361559730676146461246, 8.859931213900179330269124337239, 9.458480175949074480880339944139, 10.25403509133705822678716791876